Mindfulness requires an interaction between the past, present and future:

“The Canon defines alertness (sampajanna) as knowing both events in the mind and activities of the body as they are happening:

"And how is a monk alert? There is the case where feelings are known to the monk as they arise, known as they persist, known as they subside. Thoughts are known to him as they arise, known as they persist, known as they subside. Perceptions are known to him as they arise, known as they persist, known as they subside. This is how a monk is alert.”—-SN 47:35

“Taken together, these mental factors of sati, alertness, and ardency indicate that right sati, as a factor of the path to the end of suffering and stress, brings memories from the past to bear on a clear alertness of events and action in the present with the purpose of abandoning unskillful qualities and developing skillful qualities in the present and on into the future. In this way, alertness and ardency ensure that right sati points not only in one direction, to the past, but to all three directions of time at once: past, present and future.”
[…]
“The Buddha, in including right mindfulness in the path, takes the role that mindfulness plays in any experience where memory is brought to bear on the present and points it in a skillful direction. This is an important point to note. Instead of telling you to abandon past memories so as to approach the present with totally fresh eyes and bare awareness, he’s saying to be selective in calling on the apropriate memories that will keep you on the path to the end of suffering. And instead of telling you to watch passively as things arise and pass away on their own, he’s saying to keep remembering the need to complete any uncompleted tasks required by the path, and to protect any attainments that have already been attained. In other words, there are some things you have to remember to make arise and to prevent from passing away. (AN 4:245)—-“Right Mindfulness”, Thanissaro.

the term "memory" has not been explicitly mentioned or included in the teachings.

There are three levels of the mind according to teaching.
Latent, transgression and manifestation.

Can you explain where this classification comes from? Which teaching?

Mike

Root condition. Buddhist training is directed towards eliminating the defilements (kilesā). The
foremost defilements are the three unwholesome roots—greed, hate, and delusion. From these
spring others: Conceit (māna), speculative views (diṭṭhi), sceptical doubt (vicikicchā), mental
torpor (thīna), restlessness (uddhacca), shamelessness (ahirika), lack of moral fear or conscience
(anottappa). These defilements function at three levels:
1) Transgression (vītikkama) leading to evil bodily and verbal acts. This is checked by the
practice of morality, observing the five precepts.
2) Obsession (pariyuṭṭhāna) when the defilements come to the conscious level and
threaten to lead to transgression if not restrained by the practice of mindfulness.
3) Latency (anusaya) where they remain as tendencies ready to surface through the
impact of sensory stimuli. Security from the defilements can be obtained only by
destroying the three roots—greed, hate and delusion—at the level of latency. This
requires insight-wisdom (vipassanā-paññā), the decisive liberating factor in Buddhism.

The word sati relates to memory.
And what is the faculty of mindfulness? There is the case where a monk, a disciple of the noble ones, is mindful, highly meticulous, remembering & able to call to mind even things that were done & said long ago. SN 48.10
More interesting answers found here: https://buddhism.stackexchange.com/ques ... -awareness

16. 'In what has the world originated?' — so said the Yakkha Hemavata, — 'with what is the world intimate? by what is the world afflicted, after having grasped at what?' (167)

17. 'In six the world has originated, O Hemavata,' — so said Bhagavat, — 'with six it is intimate, by six the world is afflicted, after having grasped at six.' (168)

the term "memory" has not been explicitly mentioned or included in the teachings.

There are three levels of the mind according to teaching.
Latent, transgression and manifestation.

The three levels of the mind above seems to be three levels of defilements. B. Bodhi wrote in his book, the 8 fold path, defilements are stratified into three layers, latent, manifestation and trangression. Mind=defilements?

Whatever knowledge / experience one accumulated became memory .
Unless , one is without any knowledge ,
the mind is blank , without going through any experience then there is no memory retain whatsoever .
Thinking , pondering , any mind activities is the process of interaction and exchanging of knowledge and experience which is memory .

Is there any sutta that directly use the term "living in the present moment"?

You shouldn't chase after the past or place expectations on the future. What is past is left behind. The future is as yet unreached. Whatever quality is present you clearly see right there, right there. Not taken in, unshaken, that's how you develop the heart. Ardently doing what should be done today, for — who knows? — tomorrow death. There is no bargaining with Mortality & his mighty horde. Whoever lives thus ardently, relentlessly both day & night, has truly had an auspicious day: so says the Peaceful Sage.

The word sati relates to memory.
And what is the faculty of mindfulness? There is the case where a monk, a disciple of the noble ones, is mindful, highly meticulous, remembering & able to call to mind even things that were done & said long ago. SN 48.10
More interesting answers found here: https://buddhism.stackexchange.com/ques ... -awareness